
We apply Theorem 19 in [1]} to bound the bracketed term under the hypothesis \(|A|^2 |X| = O(p^2)\) . In the notation of [1]}, we apply Theorem 19 (or the discussion above it) to \(B=X\)  and \(C=A\) . So \(P = A \times X\)  and hence \(|L|= |A| |X|\) . We note that \(|L| |A| = |A|^2 |X| = O(p^2)\) . Therefore the number of solutions to \(x_1(a_1-a_2) = x_2(a_3-a_4)\)  with the \(a_i \in A\)  and \(x_j \in X\)  is
\(O(|A|^3 |X|^{3/2} + |A|^2 |X| (|A| + |X|)) = O(|A|^3 |X|^{3/2} + |A|^2 |X|^2).\) 
